A World Beta

Published in Investing on 8 March 2010

The Capital Asset Pricing Model is a gross oversimplification based on ludicrous assumptions.

A scientist walks into a bar. He orders a pint and then announces, in a Doc-Brown-from-Back-to-the-Future sort of way, that he can predict the behaviour of every single person in the room. The pub falls silent and turns to stare; tough miners break off their half-finished arm wrestling matches; gruff old men set down dimpled glasses onto soggy beermats; the landlord wishes he'd hired bouncers for the night.

Got Your Attention?

The scientist has everyone's attention, so he carries on. He reckons that people's actions are the product of their relationship with a man who doesn't really exist. This man is a sort of super-strength Mr. Average, and he goes by the name of Martin Portfolio.

Gesturing around the bar, the scientist claims that all people have a level of sensitivity to what Martin Portfolio does. And the size of that sensitivity is the only factor that accounts for their actions.

"What's the sensitivity called, then?" cries a voice from over by the dartboard.

"Beta!" The scientist whirls his manic eyes from table to table. "It's all about your Beta!"

The publican knows his duty: "I think you've had enough, pal. Hop it!"

Several big patrons take their cue from the landlord and manhandle this raving rationalist towards the exit. On the way out he babbles something about linearity; something else about infinite borrowing and lending at the risk-free rate; as the doors slam shut drinkers swear he screams that all people are perfectly rational and risk-averse…

William Sharpe

Mr. Sharpe is to blame, he of Sharpe Ratio fame. He came up with this little beauty:

Rp=Rf+β(Rm-Rf)

  • 'Rp' is the return of your investment. This is the thing that CAPM attempts to estimate: "what should my reward be for holding this share, this fund, or this index."

  • 'Rf' is the risk-free rate: a zero volatility investment.

  • 'Rm' is the return on something called the 'Market Portfolio'.

  • Lastly, 'β' (Beta) is the sensitivity of your investment (your 'Rp') to the 'Market Portfolio'.

It's a basic linear equation: 'linear' because the relationship it describes is a straight line instead of curved.

Assumptions

Sharpe laid down six assumptions for his equation to hold true:

1. All investors make rational decisions all of the time, based on knowledge of risk and return alone.

2. All investors have the same holding period.

3. There is no single buyer or seller large enough to move the market price of an asset

4. There are no taxes or transaction costs (markets are 'frictionless').

5. Information is free and simultaneously available to all investors.

6. Everyone is able to borrow and lend infinite amounts at the risk free rate.

Are they valid? Let's take them one at a time and see:

1. I'm not a robot; neither are you. Do you know any perfectly rational people?

2. Not true. My pension is on a 30+ year time horizon. What about yours?

3. Tell that to Goldman Sachs.

4. Dream on.

5. Tell that to Rupert Murdoch and Google.

6. I wish.

'Market Portfolio'

The 'Market Portfolio' is supposed to represent every single investable asset in the world. Shares, bonds, property, rare Batman comics, the lot. The idea of CAPM is that every individual investment has a sensitivity to the Market Portfolio, represented by Beta.

So a unit trust with a Beta of one, for example, should be expected to return exactly as much as the Market Portfolio.

Higher Beta equals higher returns: a fund with a Beta of 1.5 should offer you more than the Market.

CAPM is where we get the idea of 'Alpha' from. Let's try an example: if I know that the Risk Free Rate is 3%, the Market Portfolio will give me 10% and the Beta of my investment is 1.4, I can plug these numbers into the CAPM equation and calculate the return I should expect.

Rp=Rf+β(Rm-Rf)

Rp=3+1.4(10-3)

Rp=3+9.8=12.8%

So, thanks to CAPM, I can estimate the return that I should earn for holding something that is more volatile (higher Beta) than the Market Portfolio.

When attempting to value a share, you can plug the results from CAPM into the 'required return' part of the Gordon Growth Model. Higher required returns equal lower 'fair' values.

Alpha

If CAPM says that I should receive 12.8% reward for tolerating Beta of 1.4 we can go a stage further and think about fund manager 'skill'. This is not the place to reopen the 'active/passive' battle! Let's agree a suspension of hostilities on that one and assume, for a second, that some fund managers have a magic quality called 'skill'.

If the fund in our example actually went on to earn 15% for investors, and all other factors remained unchanged, then there would be a 2.2% discrepancy. This can be interpreted as extra return earned by the manager without (and this is the crucial bit) taking extra risk. The fund has outperformed its CAPM estimate by 2.2%, so we would describe it as having an Alpha of 2.2%.

Alpha can't always be trusted, though. CAPM tells us that the only thing that matters is volatility in relation to the Market Portfolio. But what about assets that are risky, but not volatile? Commercial property and some hedge funds fall into this category. They can appear to have extremely high alphas at times because of the way the investments are priced; it doesn't necessarily follow that the managers are highly skilled.

A Tough Job

CAPM tries to reduce the actions of millions of buyers and sellers, with an infinite number of motivations, to one tiny little formula. That's some challenge. 

Does it work? No. 

Academics have tested CAPM against real market returns many times; there has never been any relationship found between actual returns and CAPM-implied returns.

But CAPM does solve a language problem: the market is such a complex and unpredictable place that we need some sort of framework to order our discussions about risk and reward. That's what CAPM gives us, and until somebody has a better idea we are stuck with it, problems and all.

More from R J Johnson:

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Comments

The opinions expressed here are those of the individual writers and are not representative of The Motley Fool. If you spot any comments that are unsuitable hit the flag to alert our moderators.

Terrapin1 08 Mar 2010 , 9:07am

Here's a real simple formula:
Markets are as bent as a nine bob note.
So the Fool's mantra of buy and hold will work in selective periods with a very long time horizon.
Trading is a matter of discipline.
In summary GS=666

BarrenFluffit 08 Mar 2010 , 10:26am

lol; interesting explanation of where CAPM sits;

thebuffoon 08 Mar 2010 , 4:18pm

Why would you think that an article about something that is a gross oversimplification, and based on ludicrous assumptions is worth writing, let alone someone else reading?

dickdom 02 Nov 2010 , 8:58pm

I suspect it is the same as making assumptions for a mathematical or scientific model. You attempt to get an approximation, while making assumptions, then try to work out how reality differs in any specific case and what changes in the anwer would result.

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